Simpson has suggested that some of the hostility to f.o.m. in the mathematical
community may be a result of increasing compartmentalization and
specialization of mathematics, so that research which tends to treat
mathematics as a unified whole (or more generally any attempt to breach the
boundaries between areas of mathematics) is viewed as an encroachment.
I'm not sure this is the explanation of most hostility to f.o.m. (last month I
posted my own attempt at explaining some of it), but the trend towards
compartmentalization is troubling in its own right.
It is often said that the last "universal mathematicians" were Poincare and
Hilbert who were in their prime a century ago, and the reason there are no
such "polymaths" nowadays is that mathematics itself has grown too big. I
don't believe this reason, because in my opinion the increasing logical unity
of mathematics ought to counterbalance this growth. But it is true that
practically all the great mathematicians of this century have been more
specialized than Poincare and Hilbert were.
A few months ago I asked if anyone could come up with names of modern
mathematicians who have made major contributions to more than two areas of
mathematics. I've thought of three: Von Neumann (Algebra, foundations,
mathematical physics, computation theory, game theory), Kolmogorov
(probability, computation theory, dynamical systems), and Conway (knot theory,
finite groups, game theory). There ought to be more!
Why don't more contemporary mathematicians have such breadth? Even if
mathematics is too big for anyone to expect to know all of it, it certainly
ought to be possible to work in more than one area (even two is rare). My
guess is the reasons for today's specialization of mathematicians are
sociological and psychological rather than intrinsic to mathematics. I have
some arguments to support this guess, but I'd like to hear what others on FOM
have to say about this.
-- Joe Shipman